From the elliptic regulator to exotic relations

From the elliptic regulator to exotic relations

Nouressadat TOUAFEK

Abstract

In this paper we prove an identity between the elliptic regulators of some 2-isogenous elliptic curves. This allow us to prove a new exotic relation for the elliptic curve 20A of Cremona’s tables. Also we prove the (conjectured) exotic relation for the curve 20B given by Bloch and Grayson in [3].

1 Introduction

For some elliptic curves, the elliptic dilogarithm satisfies linear relations, called exotic by Bloch and Grayson [3].

In [3] a list of elliptic curves that satisfies exotic relations is given. Recently some of these relations are proved by Bertin [1], Touafek [7].

We note that whenever we can find a tempered model of the elliptic curve, the existence of exotic relations is related to elements in the second group of the K-theory K₂(E) hence to elliptic regulators, so we can prove the exotic relations.

Bloch and Grayson conjectured the following fact.

Conjecture 1 Suppose that E(Q)_tors is cyclic and d = #E(Q)_tors > 2.

Write Σ for the number of fibres of type I_ν with ν ≥3 in the N´eron model, and suppose_d−1

2

−Σ>1. Then there should be at least_d−1

2

−Σ−1exotic relations

[^d−1₂ ]

r=1

a_rD^E(rP) = 0,

Key Words: Elliptic curves, Elliptic dilogarithm, Elliptic regulator, Exotic relations.

Mathematics Subject Classification: 11G05, 14G05, 14H52.

Received: April, 2008 Accepted: September, 2008

117

where P is ad-torsion point and a_r∈Z.

In particular, these conditions are satisfied for the curve 20Aof Cremona’s tables [4]: E(Q)_tors cyclic,d= 6 and Σ = 0.

In section 3 we use an identity between regulators and some equalities between the elliptic dilogarithm to prove a new exotic relation for the elliptic curve 20A and also we prove the (conjectured) exotic relation for the curve 20B of Cremona’s tables given by Bloch and Grayson in [3].

2 Preliminaries

LetE be an elliptic curve defined overQ.

Throughout this paper, the notationE= [a₁, a₂, a₃, a₄, a₆] means that the elliptic curveE is in the Weierstrass form

y²+a₁xy+a₃y=x³+a₂x²+a₄x+a₆. 2.1 The elliptic regulator

Let F be a field. By Matsumoto’s theorem, the second group of K-theory K₂(F) can be described in terms of symbols{f, g}, forf, g∈F^∗and relations between them.

The relations are

⎧⎨

⎩

{f1f₂, g} ={f1, g}+{f2, g}

{f, g1g₂} ={f, g1}+{f, g2} {1−f, f} = 0.

For example, if v is a discrete valuation on F with maximal ideal M and residual fieldk, the Tate’s tame symbol

(x, y)_ν≡(−1)^ν(x)ν(y)x^ν(y)

y^ν(x) modM defines a homomorphism

λ_v:K₂(F)−→k^∗.

Let Q(E) be the rational function field of the elliptic curve E. To any P ∈ E(Q) is associated a valuation on Q(E) that gives the homomorphism

λ_P :K₂(Q(E))−→Q(P)^∗

and the exact sequence

0→K₂(E)⊗Q→K₂(Q(E))⊗Q→

P∈E(Q)

Q(P)^∗⊗Q→...

By definitionK₂(E) is defined modulo torsion by K₂(E)kerλ= ∩

Pkerλ_P ⊂K₂(Q(E)) where

λ:K₂(Q(E))⊗Q−→

P∈E(Q)

(Q(P)^∗⊗Q).

Definition 1 A polynomial in two variables is tempered if the polynomial of the faces of its Newton polygon has only roots of unity.

When drawing the convex hull of points (i, j) ∈ Z² corresponding to the monomialsa_i,jxⁱy^j, a_i,j = 0, you also draw points located on the faces. The polynomial of the face is a polynomial in one variabletwhich is a combination of the monomials 1,t,t²,.... The coefficients of the combination are given when going along the face, that isa_i,j if the lattice point of the face belongs to the convex hull and 0 otherwise.

In particular, the polynomials

P₁(X₁, Y₁) =Y₁²+ 2X₁Y₁−X₁³+X₁ and

P₂(X₂, Y₂) =Y₂²+ 2X₂Y₂+ 2Y₂−(X₂−1)³

are tempered, so we get {X₁, Y₁} ∈ K₂(E₁) and {X₂, Y₂} ∈ K₂(E₂), see Rodriguez-Villegas [5]. HereE₁is the elliptic curve defined byP₁(X₁, Y₁) = 0 andE₂ the elliptic curve defined byP₂(X₂, Y₂) = 0.

Letf and gbe inQ(E)^∗. Let us define

η(f, g) = log|f|d(argg)−log|g|d(argf).

Definition 2 The elliptic regulatorrof E is given by r: K₂(E)−→ R

{f, g} −→ _2π¹

γη(f, g)

for a suitable loop γ generating the subgroup H₁(E,Z)⁻ of H₁(E,Z), where the complex conjugation acts by−1.

2.2 The elliptic dilogarithm We have two representations forE(C)

E(C)−→^∼ C/(Z+τZ) −→^∼ C^∗ q^Z (℘(u), ℘^´(u))−→ u( mod Λ)−→ z=e^2πiu

where ℘is the Weierstrass function, Λ = {1, τ} the lattice associated to the elliptic curve andq=e^2πiτ.

Definition 3 The elliptic dilogarithm D^E [2] is defined by

D^E(P) =

n=+∞

n=−∞

D(qⁿz),

where P ∈E(C) is the image of z∈C^∗,z =e^2πiu, u=ξτ+η and D is the Bloch-Wigner dilogarithm

D(x) :=Li₂(x) + log|x|arg(1−x).

Remark 1 1. The Bloch-Wigner dilogarithm is a function univalued, real analytic in P¹(C)\{0,1,∞}, continuous in P¹(C)[9].

2. There is a second representation of the elliptic dilogarithm given by Bloch [2], [10] in terms of Eisenstein-Kronecker series

D^E(P) =(τ)²

π (

m,n∈Z (m,n)=(0,0)

exp( 2πi(nξ−mη)) (mτ+n)²(m⁻τ+n)

). (1)

3. The elliptic dilogarithm can be extended to divisors onE(C) D^E((f)) =

i

n_iD^E(P_i), where

(f) =

i

n_i[P_i].

2.3 The diamond operation

LetZ[E(C)]⁻ be the subgroup ofZ[E(C)] modulo the equivalence relation cl([−P]) =−cl([P]).

Definition 4 The diamond operation is defined by

♦ : Z[E(C)]× Z[E(C)] −→ Z[E(C)]⁻

((f),(g)) −→

i,jn_im_jcl([P_i−P_j]) where

(f) =

i

n_i[P_i] and(g) =

j

m_j[P_j].

The following theorem [2], establishes the relation between the elliptic regula- tor and the elliptic dilogarithm.

Theorem 1 The elliptic dilogarithm D^E can be extended to a morphism Z[E(C)]⁻−→R.

If f, g are functions on E and{f, g} ∈K₂(E), then πr({f, g}) =D^E((f)♦(g));

in particular

D^E((f)♦(1−f)) = 0.

3 An identity between regulators

LetE₁ be the elliptic curve, isomorphic to the curve 20A,with equation Y₁²+ 2X₁Y₁=X₁³−X₁

and letE₂ be the elliptic curve, isomorphic to the curve 20B, with equation Y₂²+ 2X₂Y₂+ 2Y₂= (X₂−1)³.

Proposition 1 We have

πr({X1, Y₁}) = −4D^E1(P₁)−4D^E1(2P₁) πr({X2, Y₂}) = 6D^E2(P₂) + 6D^E2(2P₂)

where P₁= (−1,2)is the 6-torsion point of the curve E₁= [2,0,0,−1,0]and P₂= (5,4)is the 6-torsion point of the curveE₂= [2,−3,2,3,−1].

Proof. We need to compute the following divisors (X₁) = 2 [3P₁]−2 [O₁]

(Y₁) = [3P₁] + [4P₁] + [5P₁]−3 [O₁] (X₂) = 2 [3P₂]−2 [O₂] (Y₂) = 3 [2P₂]−3 [O₂]

hence

(X₁)♦(Y1) =−4cl([P1])−4cl([2P₁]) and

(X₂)♦(Y₂) = 6cl([P₂]) + 6cl([2P₂]) so, by theorem 1 we get

πr({X₁, Y₁}) =−4D^E1(P₁)−4D^E1(2P₁) and

πr({X₂, Y₂}) = 6D^E2(P₂) + 6D^E2(2P₂).

Remark 2 Using the previous proposition and formula (1), we have find by the computer,

r({X₂, Y₂})=^? r({X₁, Y₁}),

where the notationA=^? B, means ” A is conjectured to be equal to B ”, that is A and B are numerically equal to at least 25 decimal places.

Theorem 2 We have the following identity

r({X₂, Y₂}) =r({X₁, Y₁}).

Proof. Let Ξ¹= 20Abe the elliptic curve with equation S₁²=T₁³+T₁²−T₁

and Ξ² be the elliptic curve, isomorphic to 20B, with equation S₂²=T₂³−2T₂²+ 5T₂.

It is easy to check that

T₁=X₁, S₁=Y₁+X₁ (2) and

T₂=X₂, S₂=Y₂+X₂+ 1 (3) give isomorphisms

Ξ¹E₁, Ξ²E₂. Also we have the 2-isogeny [6] given by

Φ : Ξ¹ −→Ξ² (T₁, S₁) −→

S₁²

T₁²,−^S1(T_T¹²2⁺¹⁾ 1

. (4)

Using (2), (3) and (4) we can see Φ as

Φ : E₁ −→E₂

(X₁, Y₁) −→(X^Φ, Y^Φ) = (^(Y1+X_X2¹⁾²

1 ,−^(Y1+X1+1)(Y_X2^1+X12+X1)

1 ).

The elliptic curveE₁ can be considered as a double cover ofP¹ by π_X1 :E₁−→P¹

ramified at the zeros ofX₁³+X₁²−X1,i.e. 0,⁻¹⁺₂^√5,⁻¹⁻₂^√5.The closed curve σ₁=π⁻¹_X1([0,⁻¹⁺₂^√5]) generatesH₁(E₁,Z)⁻.

The elliptic curveE₂ can be considered as a double cover ofP¹ by π_X2 :E₂−→P¹

ramified at the zeros ofX₂³−2X₂²+ 5X₂,i.e. 0,1 + 2i,1−2i.The closed curve σ₂=π⁻¹_X2([1−2i,1 + 2i]) generatesH₁(E₂,Z)⁻.

Using the 2-isogeny we get

2P₁, 5P₁ −→Φ 2P₂ P, Q −→Φ 3P₂ P₁, 4P₁ −→Φ 4P₂ 3P₁, O₁ −→Φ O₂ where

P = (−ϕ, ϕ), Q= (−1 ϕ,1

ϕ), Q=P+ 3P₁, ϕ= 1 +√ 5 2 .

Also, when (X₂, Y₂) describesσ₂, (X^Φ, Y^Φ) describes twice the closed curve σ={(X₁, Y₁)∈E₁(C)/|X₁|= 1},

which generatesH₁(E₁,Z)⁻, because it’s in the same homology class asσ₁. Hence,

r({X₂, Y₂}) =±1 2r

X^Φ, Y^Φ

. (5)

We have

(1 +X₁+Y₁) = [2P₁] + 2 [5P₁]−3 [O₁] (X₁+Y₁) = [3P₁] + [P] + [Q]−3 [O₁] (Y₁+X₁+X₁²) = [3P₁] + [5P₁] + 2[2P₁]−4 [O₁].

Performing the necessary computation, we obtain

(X^Φ)♦(Y^Φ) = −12cl([P1]) + 12cl([2P₁]) + 12cl([P−2P₁]) + 12cl([P−2Q]) +12cl([Q−2P₁]) + 12cl([Q−2P])

and

(1 +X₁+Y₁)♦(X₁+Y₁) = 5cl([P₁])−cl([2P₁])−3cl([P−2P₁])−3cl([P−2Q])

−3cl([Q−2P₁])−3cl([Q−2P]).

Using the fact that

D^E((1 +X₁+Y₁)♦(X₁+Y₁)) = 0 we get

D^E1(X^Φ♦Y^Φ) = 8D^E1(P₁) + 8D^E1(2P₁) so by Theorem 1 and Proposition 1

r

X^Φ, Y^Φ

=−2r({X1, Y₁}). (6) By (5), (6) and remark 2 we get

r({X2, Y₂}) =r({X1, Y₁}).

LetE₁,E₂ be as above. We have the following theorem [8].

Theorem 3 We have the following equalities

1) D^E1(P₁) =−2D^E2(P₂) + 3D^E2(2P₂) 2) D^E1(2P₁) =−2D^E2(P₂) + 2D^E2(2P₂).

Proof. The proof follow the same way of the proof of Theorem 3.2 in [7]

Now, we are able to give a new exotic relation for the curve 20A.

Corollary 1 We have the linear relation

16D^E1(P₁)−11D^E1(2P₁) = 0.

Proof. It results from Proposition 1 and Theorem 2 that

−4D^E1(P₁)−4D^E1(2P₁) = 6D^E2(P₂) + 6D^E2(2P₂);

so by theorem 3, we get

16D^E1(P₁)−11D^E1(2P₁) = 0.

Remark 3 1. In turn, by Theorem 3, the relation 16D^E1(P₁)−11D^E1(2P₁) = 0 becomes

5D^E2(P₂)−13D^E2(2P₂) = 0.

This achieves the proof of the (conjectured) exotic relation for the curve 20B given by Bloch and Grayson in [3].

2. In [3] only elliptic curves with negative discriminant are considered, so our new exotic relation does not appear in the list of Bloch and Grayson because the curve20A have a positive discriminant.

References

[1] M.J. Bertin, Mesure de Mahler et r´egulateur elliptique: Preuve de deux relations exotiques, CRM Proc. Lectures Notes,36(2004), 1-12.

[2] S. Bloch,Higher regulators, algebraic K-theory and zeta functions of ellptic curves, ( Irvine Lecture, 1977) CRM-Monograph series, vol.11, Amer.Math.Soc., Providence, RI, 2000.

[3] S. Bloch and D. Grayson,K2andL-functions of elliptic curves computer calculations, I, Contemp. Math.55(1986), 79-88.

[4] J.E. Cremona, Algorithms for modular elliptic curves, Cambridge University Press, 2nd edition, 1997.

[5] F. Rodriguez-Villegas,Modular Mahler measures I, Topics in Number Theory ( S. D.

Ahlgren, G. E. Andrews, and K. Ono, eds), Kluwer, Dordrecht, 1999, pp. 17-48.

[6] J.H. Silverman, The arithmetic of elliptic curves, Springer Verlag, Berlin and New York, 1986.

[7] N. Touafek,Some equalities between elliptic dilogarithm of 2-isogenous elliptic curves, Int. J. Algebra,2(2008), 45-51.

[8] N. Touafek, Th`ese de Doctorat, Universit´e de Constantine, Algeria, 2008.

[9] D. Zagier, Dedekind Zeta functions, and the Algebraic K-theory of Fields, Arith- metic algebraic geometry (Texel, 1989), 391-430, Progr. Math., 89, Birkhuser Boston, Boston, MA, 1991.

[10] D. Zagier and H. Gangl, Classical and elliptic polylogarithms and special values of L-series, the Arithmetic and Geometry of Algebraic cycles, Nato, Adv. Sci. Ser.C.

Math.Phys.Sci., Vol. 548, Kluwer, Dordrech, 2000, pp. 561-615.

Laboratoire de Physique Th´eorique Equipe de Th´eorie des Nombres Universit´e de Jijel, Algeria [email protected]